Buildings, streets, and entire cities crash into one another. Countless urban details – housing block windows, city maps – overwhelm your field of vision. Black squiggles race across the surface surrounded by flashes of colour: a beam of yellow, a red parallelogram.

The world Julie Mehretu paints is bogglingly chaotic. Yet when I meet the American artist in her light-soaked workspace overlooking New York’s Hudson River (Martha Stewart has her office a few storeys below), the mood in the studio is the exact opposite: calm, collected, in total control. She is preparing for not one but two major solo exhibitions – one in New York, one in London – and a pair of assistants quietly help her apply the final painstaking touches to her latest paintings. Bird’s-eye views of cities in the Middle East are taped to the walls. The bookshelves groan with volumes on everything from French genre painting to Ethiopian history; her studio is as much a think tank as a laboratory. And Mehretu’s wife, the artist Jessica Rankin, has stopped by – they’re preparing to leave town, and they have two sons’ schedules to get in order.

Motherhood, Mehretu tells me, has led her to adopt a rigorous daily routine: in the studio by 8.30am, back home in Harlem around 6. “I’m a mom! When you’re not a mom you can get up in the middle of the night, paint, sleep all morning ... you can’t do that when you have two children!”

Hanging all around us are new paintings, freer and messier than her early work, that confirm an impression that’s been building for years: Mehretu, without playing the art world’s exhausting social games, has become the most significant American painter of her generation. (She’s already one of the priciest; at Christie’s in New York last month a painting of hers sold for a record $4.6m.) They combine rigorously drawn architectural and urban details with more gestural, even aggressive brushwork. “You have all these layers that have fused together,” she tells me, “and they become part of one larger machine.”

Wandering in her studio I come across a table strewn with folders of photographs and maps. The labels offer a key to her new work: Baghdad. Tahrir Square. Pakistan. The floods of information in her art reflect the contemporary world, but there’s always too much going on to make everything out. “I’m interested in the experience you can have while looking ... information that coalesces into something that comes out of the painting towards you. The amount of time that goes into these is very apparent, and that can be overwhelming. It becomes a physical experience.”

Mehretu was born in Addis Ababa in 1970 to an Ethiopian father and an American mother. When the military junta known as the Derg began a campaign of terror in 1977, her family fled to the United States. “Coming from this African background, you’re the children of people who were there during decolonisation, when the world really fundamentally shifted and this other form of modernism emerged,” Mehretu explains. “Now we’re all dislocated ... and there’s this constant negotiating of place, space, ideals, ideas.” Her childhood on two continents, and her long engagement with postcolonial writers and artists, are always just below the surface of her work. “I am only the person I am because I was raised in this. But you can’t think this way when you’re making a painting. You can only think this way when you’re finished making a painting.”

She first came to prominence in the early 2000s for her lustrous, dynamic, abstract paintings, usually drawing on architectural imagery and sometimes done on multiple sheets of translucent vellum that she layered one on top of the other. She was celebrated for giving a new life to abstraction, with shows everywhere from the Guggenheim in New York to the Louisiana Museum in Denmark. She also won a massive commission from an unlikely art patron: Goldman Sachs, for whom she produced a turbulent 80-foot mural inspired by the history (and troubles) of capitalism.

Yet for all their ambition, her earlier work could feel safe at times – too thought through, perhaps, or too streamlined. These new paintings take more chances; they’re the work of an artist at the top of her game. Each canvas is awash with architectural particulars such as building facades and grids of city streets, overlain until they become a torrent of unintelligible lines and shapes. Then they’re interlaced with swooping marks that suggest migration, or the noise of digital communication. Mehretu’s paintings come closer than any other artist’s today to conveying the world we now live in: culturally hybrid and in constant motion, a synthesis of skyscrapers and ruins.

“I don’t ever work in a way where something is an illustration of an event, but when something is occurring at the same time I see it as very informed by that,” Mehretu tells me. Invisible Line, first seen in Venice and now on view at White Cube in London, began in 2010 as a response to the architecture of New York – but when the Egyptian revolution began to flourish her work started to change. “I was in here working on New York, and I’m drawing, and this thing is unfolding: I have al-Jazeera on the computer livestream, I’m paying attention to NPR … So I was looking architecturally at New York, and then suddenly I’m back in Africa. And then the painting grows through drawing after drawing, layer after layer.”

You can see those same themes of global upheaval in Mogamma (A Painting in Four Parts) – Mogamma, the name of Cairo’s main government building, means “collective” in Arabic – also on show in London. Images come together not just from Egypt, but from cities worldwide where public squares have formed the backdrop for change, not just now but in centuries past. Tahrir bleeds into Tiananmen, the Place de la Bastille, and even Zuccotti Park – the tiny Manhattan square where Occupy Wall Street exploded. “I was thinking of the events that happened not just in Cairo and Tunis, not just the Arab spring, but spread everywhere. There were the protests in Wisconsin, you had the Occupy movement in New York. It was an explosion, all these little fires that lit of everywhere. I was really interested in that gesture of the occupation of a public square: what did that mean? How did that operate?”

The ongoing political transformation in Egypt continues to drive Mehretu’s thinking in the studio. Last summer Mehretu travelled to Cairo, a few days after Mohamed Morsi was sworn in as president, and stood in the massive square where a popular movement had brought down a dictator she knew all her life. When she came back she painted Beloved (Cairo), the 24-foot anchor painting of her exhibition at Marian Goodman Gallery in New York. Tahrir Square stands at the center of a maelstrom of overlapping buildings and streets, soaring curves, and indecipherable marks and lines: the square seems to be generating unstoppable energy, to be harnessed for good or ill.

“I’d been following the situation so closely, and I was asking myself, what was this moment?” she says. “There’s no way the gesture of Tahrir Square was futile. But history goes in fits and starts. There’s this tension and contradiction, this weird entropic cycle of utopian ideals and the impossibility of that. That’s what I’m interested in: the space in between, the moment of imagining what is possible and yet not knowing what that is.”

She paints slowly – some works take a year to produce – and she wants her paintings viewed slowly too. If that means that not every viewer will spend the time working through her immensely detailed work, Julie Mehretu says she doesn’t mind. “All art chooses its own audience,” she says. “People look at film in a gallery, and if they walk out after two minutes they know they haven’t seen the whole work. But then people look at a painting for two minutes and think they’ve seen it. Certain paintings are made to be consumed fast. But some require a slowed-down time. You have to go back to them.”